Update Data Structures Class Template.cpp

Data Structures Class Template.cpp

@@ -5827,6 +5827,108 @@ pair<ll,int> weight_blossom()
 }
 //End Max Weight Matching
 
+//Linear Recurrence
+#define SZ 233333
+const int MOD=1e9+7; //or any prime
+ll qp(ll a,ll b)
+{
+	ll x=1; a%=MOD;
+	while(b)
+	{
+		if(b&1) x=x*a%MOD;
+		a=a*a%MOD; b>>=1;
+	}
+	return x;
+}
+namespace linear_seq {
+inline vector<int> BM(vector<int> x)
+{
+	//ls: (shortest) relation sequence (after filling zeroes) so far
+	//cur: current relation sequence
+	vector<int> ls,cur;
+	//lf: the position of ls (t')
+	//ld: delta of ls (v')
+	int lf,ld;
+	for(int i=0;i<int(x.size());++i)
+	{
+		ll t=0;
+		//evaluate at position i
+		for(int j=0;j<int(cur.size());++j)
+			t=(t+x[i-j-1]*(ll)cur[j])%MOD;
+		if((t-x[i])%MOD==0) continue; //good so far
+		//first non-zero position
+		if(!cur.size())
+		{
+			cur.resize(i+1);
+			lf=i; ld=(t-x[i])%MOD;
+			continue;
+		}
+		//cur=cur-c/ld*(x[i]-t)
+		ll k=-(x[i]-t)*qp(ld,MOD-2)%MOD/*1/ld*/;
+		vector<int> c(i-lf-1); //add zeroes in front
+		c.pb(k);
+		for(int j=0;j<int(ls.size());++j)
+			c.pb(-ls[j]*k%MOD);
+		if(c.size()<cur.size()) c.resize(cur.size());
+		for(int j=0;j<int(cur.size());++j)
+			c[j]=(c[j]+cur[j])%MOD;
+		//if cur is better than ls, change ls to cur
+		if(i-lf+(int)ls.size()>=(int)cur.size())
+			ls=cur,lf=i,ld=(t-x[i])%MOD;
+		cur=c;
+	}
+	for(int i=0;i<int(cur.size());++i)
+		cur[i]=(cur[i]%MOD+MOD)%MOD;
+	return cur;
+}
+int m; //length of recurrence
+//a: first terms
+//h: relation
+ll a[SZ],h[SZ],t_[SZ],s[SZ],t[SZ];
+//calculate p*q mod f
+inline void mull(ll*p,ll*q)
+{
+	for(int i=0;i<m+m;++i) t_[i]=0;
+	for(int i=0;i<m;++i) if(p[i])
+		for(int j=0;j<m;++j)
+			t_[i+j]=(t_[i+j]+p[i]*q[j])%MOD;
+	for(int i=m+m-1;i>=m;--i) if(t_[i])
+		//miuns t_[i]x^{i-m}(x^m-\sum_{j=0}^{m-1} x^{m-j-1}h_j)
+		for(int j=m-1;~j;--j)
+			t_[i-j-1]=(t_[i-j-1]+t_[i]*h[j])%MOD;
+	for(int i=0;i<m;++i) p[i]=t_[i];
+}
+inline ll calc(ll K)
+{
+	for(int i=m;~i;--i)
+		s[i]=t[i]=0;
+	//init
+	s[0]=1; if(m!=1) t[1]=1; else t[0]=h[0];
+	//binary-exponentiation
+	while(K)
+	{
+		if(K&1) mull(s,t);
+		mull(t,t); K>>=1;
+	}
+	ll su=0;
+	for(int i=0;i<m;++i) su=(su+s[i]*a[i])%MOD;
+	return (su%MOD+MOD)%MOD;
+}
+inline int work(vector<int> x,ll n)
+{
+	if(n<int(x.size())) return x[n];
+	vector<int> v=BM(x); m=v.size(); if(!m) return 0;
+	for(int i=0;i<m;++i) h[i]=v[i],a[i]=x[i];
+	return calc(n);
+}
+}
+using linear_seq::work;
+int main()
+{
+	cout<<work({1,1,2,3,5,8,13,21},10)<<"\n";
+}
+//End Linear Recurrence
+
 int main() //Testing Zone
 {
 	ios_base::sync_with_stdio(0); cin.tie(0);